2009/04/13 by Juan Antonio Moya Pérez, Juan A. Perez, Perez, Juan A. · 1 citation
Computer Science · Mathematics · #03E30 #03E35 (Primary) #03F30 (Secondary) #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #General Mathematics (math.GM) #Mathematical and Theoretical Analysis #math.GM #msc:03E30 #msc:03E35 #msc:03F30
paper · pdf · doi:10.48550/arxiv.0904.1957
12 pages. Two references deleted. Minor editing of concluding remarks
openalex publication_date 2009/04/13 · arxiv created 2009/07/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Goodstein sequences are numerical sequences in which a natural number m, expressed as the complete normal form to a given base a, is modified by increasing the value of the base a by one unit and subtracting one unit from the resulting expression. As initially defined, the first term of the Goodstein sequence is the complete normal form of m to base 2. Goodstein's Theorem states that, for all natural numbers, the Goodstein sequence eventually terminates at zero. Goodstein's Theorem was originally proved using the well-ordered properties of transfinite ordinals. The theorem was also shown to be unprovable-in-PA (Peano Arithmetic) using transfinite induction and Godel's Second Incompleteness Theorem. This article describes a proof of Goodstein's Theorem in first-order arithmetic that contradicts the theorem's unprovability-in-PA. The proof uses mathematical induction and is applied (via the super-exponential function) to a generalized version of the Goodstein sequences. Such a proof demonstrates the inconsistency of classical set theory, more precisely the combination of the Zermelo-Fraenkel axioms and the axiom of choice (ZFC).